Matrix product operator representations

Matrix Product Operators, Matrix Product States, and ab initio Density Matrix Renormalization Group algorithms

Current descriptions of the ab initio DMRG algorithm use two superficially different languages: an older language of the renormalization group and renormalized operators, and a more recent language of matrix product states and matrix product operators. The same algorithm can appear dramatically different when written in the two different vocabularies. In this work, we carefully describe the translation between the two languages in several contexts. First, we describe how to efficiently implement the ab-initio DMRG sweep using a matrix product operator based code, and the equivalence to the ori…

A multiprecision matrix calculation library and its extension library for a matrix-product-state simulation of quantum computing

A C++ library, named ZKCM, has been developed for the purpose of multiprecision matrix calculations, which is based on the GNU MP and MPFR libraries. It is especially convenient for writing programs involving tensor-product operations, tracing-out operations, and singular-value decompositions. Its extension library, ZKCM_QC, for simulating quantum computing has been developed using the time-dependent matrix-product-state simulation method. This report gives a brief introduction to the libraries with sample programs.

Spectral Design for Matrix Hamiltonians: Different Methods of Constructing of a Matrix Intertwining Operator

We study intertwining relations for $n\times n$ matrix non-Hermitian, in general, one-dimensional Hamiltonians by $n\times n$ matrix linear differential operators with nondegenerate coefficients at $d/dx$ in the highest degree. Some methods of constructing of $n\times n$ matrix intertwining operator of the first order of general form are proposed and their interrelation is examined. As example we construct $2\times2$ matrix Hamiltonian of general form intertwined by operator of the first order with the Hamiltonian with zero matrix potential. It is shown that one can add for the final $2\times2…

Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods

Transfer matrices and matrix product operators play an ubiquitous role in the field of many body physics. This paper gives an ideosyncratic overview of applications, exact results and computational aspects of diagonalizing transfer matrices and matrix product operators. The results in this paper are a mixture of classic results, presented from the point of view of tensor networks, and of new results. Topics discussed are exact solutions of transfer matrices in equilibrium and non-equilibrium statistical physics, tensor network states, matrix product operator algebras, and numerical matrix prod…

Matrix Product States, Random Matrix Theory and the Principle of Maximum Entropy

Using random matrix techniques and the theory of Matrix Product States we show that reduced density matrices of quantum spin chains have generically maximum entropy.

Matrix Elements of Generalized Coherent Operators

Explicit forms are given of matrix elements of generalized coherent operators based on Lie algebras su(1,1) and su(2). We also give a kind of factorization formula of the associated Laguerre polynomials.

Characterisations of Matrix and Operator-Valued $Φ$-Entropies, and Operator Efron-Stein Inequalities

We derive new characterisations of the matrix $\mathrmΦ$-entropy functionals introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014]. Notably, all known equivalent characterisations of the classical $Φ$-entropies have their matrix correspondences. Next, we propose an operator-valued generalisation of the matrix $Φ$-entropy functionals, and prove their subadditivity under Löwner partial ordering. Our results demonstrate that the subadditivity of operator-valued $Φ$-entropies is equivalent to the convexity of various related functions. This result can be used to demonstrate an interesting resu…

Hermitian quasi-exactly solvable matrix Shroedinger operators

We construct six multi-parameter families of Hermitian quasi-exactly solvable matrix Schroedinger operators in one variable. The method for finding these operators relies heavily upon a special representation of the Lie algebra o(2,2) whose representation space contains an invariant finite-dimensional subspace. Besides that we give several examples of quasi-exactly solvable matrix models that have square-integrable eigenfunctions. These examples are in direct analogy with the quasi-exactly solvable scalar Schroedinger operators obtained by Turbiner and Ushveridze.

Continuous Matrix Product States for Quantum Fields

We define matrix product states in the continuum limit, without any reference to an underlying lattice parameter. This allows to extend the density matrix renormalization group and variational matrix product state formalism to quantum field theories and continuum models in 1 spatial dimension. We illustrate our procedure with the Lieb-Liniger model.

Spectral analysis of matrix scaling and operator scaling

We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent algorithm also has linear convergence under the same assumption. The spectral gap condition for operator scaling is closely related to the notion of quantum expander studied in quantum information theory. The spectral analysis also provides bounds on some important quantities of the scaling problems, such as the condition number of the scaling solution and the ca…

连续矩阵乘积算符用于量子场论

Continuous matrix product operators for quantum fields

本文提出连续矩阵乘积算符的ansatz，展示其闭式表达、连续极限及保持纠缠面积律，用于构建超越量子细胞自动机的连续矩阵乘积单元aris。

Interaction of Multiple Tensor Product Operators of the Same Type: an Introduction

Tensor product operators on finite dimensional Hilbert spaces are studied. The focus is on bilinear tensor product operators. A tensor product operator on a pair of Hilbert spaces is a maximally general bilinear operator into a target Hilbert space. By 'maximally general' is meant every bilinear operator from the same pair of spaces to any Hilbert space factors into the composition of the tensor product operator with a uniquely determined linear mapping on the target space. There are multiple distinct tensor product operators of the same type; there is no "the" tensor product. Distinctly diffe…